Type III and N Einstein spacetimes in higher dimensions: General properties

The Sachs equations governing the evolution of the optical matrix of geodetic WANDs (Weyl aligned null directions) are explicitly solved in $n$ dimensions in several cases which are of interest in potential applications. This is then used to study Einstein spacetimes of type III and N in the higher dimensional Newman-Penrose formalism, considering both Kundt and expanding (possibly twisting) solutions. In particular, the general dependence of the metric and of the Weyl tensor on an affine parameter $r$ is obtained in a closed form. This allows us to characterize the peeling behavior of the Weyl “physical” components for large values of $r$, and thus to discuss, e.g., how the presence of twist affects polarization modes, and qualitative differences between four and higher dimensions. Further, the $r$ dependence of certain nonzero scalar curvature invariants of expanding spacetimes is used to demonstrate that curvature singularities may generically be present. As an illustration, several explicit type N/III spacetimes that solve Einstein’s vacuum equations (with a possible cosmological constant) in higher dimensions are finally presented.